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Chapter 12: Q. 14 (page 953)

If the function $$f(x, y)$$ is differentiable at a point $$(a, b)$$, explain why the tangent lines to the graph of $$f$$ at $$(a, b)$$ in the $$x$$ and $$y$$ directions are sufficient to determine the tangent plane to the surface.

Short Answer

Expert verified

If the function $$f(x, y)$$ is differentiable at a point $$(a, b)$$, then the tangent lines to the graph of $$f$$ at $$(a, b)$$ in the $$x$$ and $$y$$ directions are sufficient to determine the tangent plane to the surface because they all lie in the same plane.

Step by step solution

01

Step 1. Given Information

The function $$f(x, y)$$ is differentiable at a point $$(a, b)$$

02

Step 2. Explanation

We know that the function, $$f(x, y)$$ is differentiable at a point $$(a, b)$$.

$$\implies$$ All the lines tangent to the graph of the function, $$f$$ at the point $$(a,b)$$ lie in the same plane.

Here, we can use any two distinct lines in that plane to determine the equation of the plane.

Therefore, If the function $$f(x, y)$$ is differentiable at a point $$(a, b)$$, then the tangent lines to the graph of $$f$$ at $$(a, b)$$ in the $$x$$ and $$y$$ directions are sufficient to determine the tangent plane to the surface.

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